"I was but..." meaning?

The 'but' there means "only" or "not more than". It is kind of an old way of phrasing that -- not seen too often in 20th and 21st century English...

It is the same as saying, "I was only ten years old," or, "I was just ten years old."

Using "but" is a stylistic device to stress how young the speaker was.
The expression is a way of saying
> I was nothing but ten years old.
> I was only ten years old.
> I was just ten years old.
> I was nothing more than a ten-year-old child.

Just saying "I was ten years old" does not stress the pathetic "youngness" of the speaker. It's just a factual statement.

Other uses:
> I was but a simple waitress ...
(I was only a simple waitress ... , no more than a simple waitress)

> He was but a child when his parents died.
(He was nothing but a child, only a child, nothing more than a child, just a child)

> Rover was but a dog, but I loved him.
(He was just a dog, only a dog, nothing but a little old dog)

> He was but a baby when the dingo ate him.
(Just a baby, only a baby, no more than a little baby)

It means "I was ONLY ten years old."

only, in a slightly negative or condescending sense (as in lacking something of value or being incomplete, in this case lacking age and maturity). Yet can be used in a similar but not identical sense as well (yet as in not completed or achieved as of this moment rather than yet as in still occurring). Many words can mean several different things along the same general idea.

It means only.

I means "only".

Just a manner of speech, not really intended literal.
My dad was always saying things like "when I was 13 old" etc. You could understand what he was saying just fine with an open mind. Weeks old certainly wasn't suggested in the context.
In your case, 10 years old insinuated.
Today you may expect "I was like 10 years old"